# third party imports
import numpy as np
# stdlib imports
from openquake.hazardlib.imt import PGA, PGV, SA
from shakelib.gmice.gmice import GMICE
[docs]class WGRW12(GMICE):
"""
Implements the ground motion intensity conversion equations (GMICE) of
Worden et al. (2012).
References:
Worden, C. B., Gerstenberger, M. C., Rhoades, D. A., & Wald, D. J.
(2012). Probabilistic relationships between ground‐motion parameters
and modified Mercalli intensity in California. Bulletin of the
Seismological Society of America, 102(1), 204-221.
"""
# -----------------------------------------------------------------------
#
# MMI = c2->C1 + c2->C2 * log(Y) for log(Y) <= c2->T1
# MMI = C1 + C2 * log(Y) for c2->T1 < log(Y) <= T1
# MMI = C3 + C4 * log(Y) for log(Y) > T1
#
# or
#
# MMI = c2->C1 + c2->C2 * log(Y) + C5 + C6 * log(D) + C7 * M
# for log(Y) <= c2->T1
# MMI = C1 + C2 * log(Y) + C5 + C6 * log(D) + C7 * M
# for c2->T1 < log(Y) <= T1
# MMI = C3 + C4 * log(Y) + C5 + C6 * log(D) + C7 * M for log(Y) > T1
#
# Limit the distance residuals to between 10 and 300 km.
# Limit the magnitude residuals to between M3.0 and M7.3.
#
# -----------------------------------------------------------------------
def __init__(self):
super().__init__()
self.min_max = (1.0, 10.0)
self.name = "Worden et al. (2012)"
self.scale = "scale_wgrw12.ps"
self._constants = {
self._pga: {
"C1": 1.78,
"C2": 1.55,
"C3": -1.60,
"C4": 3.70,
"C5": -0.91,
"C6": 1.02,
"C7": -0.17,
"T1": 1.57,
"T2": 4.22,
"SMMI": 0.66,
"SPGM": 0.35,
},
self._pgv: {
"C1": 3.78,
"C2": 1.47,
"C3": 2.89,
"C4": 3.16,
"C5": 0.90,
"C6": 0.00,
"C7": -0.18,
"T1": 0.53,
"T2": 4.56,
"SMMI": 0.63,
"SPGM": 0.38,
},
self._sa03: {
"C1": 1.26,
"C2": 1.69,
"C3": -4.15,
"C4": 4.14,
"C5": -1.05,
"C6": 0.60,
"C7": 0.00,
"T1": 2.21,
"T2": 4.99,
"SMMI": 0.82,
"SPGM": 0.44,
},
self._sa10: {
"C1": 2.50,
"C2": 1.51,
"C3": 0.20,
"C4": 2.90,
"C5": 2.27,
"C6": -0.49,
"C7": -0.29,
"T1": 1.65,
"T2": 4.98,
"SMMI": 0.75,
"SPGM": 0.47,
},
self._sa30: {
"C1": 3.81,
"C2": 1.17,
"C3": 1.99,
"C4": 3.01,
"C5": 1.91,
"C6": -0.57,
"C7": -0.21,
"T1": 0.99,
"T2": 4.96,
"SMMI": 0.89,
"SPGM": 0.64,
},
}
self._constants2 = {
self._pga: {"C1": 1.71, "C2": 2.08, "T1": 0.14, "T2": 2.0},
self._pgv: {"C1": 4.62, "C2": 2.17, "T1": -1.21, "T2": 2.0},
self._sa03: {"C1": 1.15, "C2": 1.92, "T1": 0.44, "T2": 2.0},
self._sa10: {"C1": 2.71, "C2": 2.17, "T1": -0.33, "T2": 2.0},
self._sa30: {"C1": 7.35, "C2": 3.45, "T1": -1.55, "T2": 2.0},
}
self.DEFINED_FOR_INTENSITY_MEASURE_TYPES = set([PGA, PGV, SA])
self.DEFINED_FOR_SA_PERIODS = set([0.3, 1.0, 3.0])
[docs] def getMIfromGM(self, amps, imt, dists=None, mag=None):
"""
Function to compute macroseismic intensity from ground-motion
intensity. Supported ground-motion IMTs are PGA, PGV and PSA
at 0.3, 1.0, and 3.0 sec periods.
Args:
amps (ndarray):
Ground motion amplitude; natural log units; g for PGA and
PSA, cm/s for PGV.
imt (OpenQuake IMT):
Type the input amps (must be one of PGA, PGV, or SA).
Supported SA periods are 0.3, 1.0, and 3.0 sec.
`[link] <http://docs.openquake.org/oq-hazardlib/master/imt.html>`
dists (ndarray):
Numpy array of distances from rupture (km).
mag (float):
Earthquake magnitude.
Returns:
ndarray of Modified Mercalli Intensity and ndarray of
dMMI / dln(amp) (i.e., the slope of the relationship at the
point in question).
""" # noqa
lfact = np.log10(np.e)
c, c2 = self._getConsts(imt)
if dists is not None and mag is not None:
doresid = True
ldd = np.log10(np.clip(dists, 10, 300))
lmm = np.clip(mag, 3.0, 7.3)
else:
doresid = False
#
# Convert (for accelerations) from ln(g) to cm/s^2
# then take the log10
#
if imt != self._pgv:
units = 981.0
else:
units = 1.0
#
# Math: log10(981 * exp(amps)) = log10(981) + log10(exp(amps))
# = log10(981) + amps * log10(e)
# For PGV, just convert ln(amp) to log10(amp) by multiplying
# by log10(e)
#
lamps = np.log10(units) + amps * lfact
mmi = np.zeros_like(amps)
dmmi_damp = np.zeros_like(amps)
#
# This is the MMI 1 to 2 range that is discussed in the paper but not
# specifically implemented there
#
idx = lamps < c2["T1"]
mmi[idx] = c2["C1"] + c2["C2"] * lamps[idx]
dmmi_damp[idx] = c2["C2"] * lfact
#
# This is the lower segment of the bi-linear fit
#
idx = (lamps >= c2["T1"]) & (lamps < c["T1"])
mmi[idx] = c["C1"] + c["C2"] * lamps[idx]
dmmi_damp[idx] = c["C2"] * lfact
#
# This is the upper segment of the bi-linear fit
#
idx = lamps >= c["T1"]
mmi[idx] = c["C3"] + c["C4"] * lamps[idx]
dmmi_damp[idx] = c["C4"] * lfact
if doresid:
mmi += c["C5"] + c["C6"] * ldd + c["C7"] * lmm
mmi = np.clip(mmi, 1.0, 10.0)
mmi[np.isnan(amps)] = np.nan
return mmi, dmmi_damp
[docs] def getGMfromMI(self, mmi, imt, dists=None, mag=None):
"""
Function to tcompute ground-motion intensity from macroseismic
intensity. Supported IMTs are PGA, PGV and PSA for 0.3, 1.0, and
3.0 sec periods.
Args:
mmi (ndarray):
Macroseismic intensity.
imt (OpenQuake IMT):
IMT of the requested ground-motions intensities (must be
one of PGA, PGV, or SA).
`[link] <http://docs.openquake.org/oq-hazardlib/master/imt.html>`
dists (ndarray):
Rupture distances (km) to the corresponding MMIs.
mag (float):
Earthquake magnitude.
Returns:
Ndarray of ground motion intensity in natural log of g for PGA
and PSA, and natural log cm/s for PGV; ndarray of dln(amp) / dMMI
(i.e., the slope of the relationship at the point in question).
""" # noqa
lfact = np.log10(np.e)
c, c2 = self._getConsts(imt)
mmi = mmi.copy()
ix_nan = np.isnan(mmi)
mmi[ix_nan] = 1.0
if dists is not None and mag is not None:
doresid = True
ldd = np.log10(np.clip(dists, 10, 300))
lmm = np.clip(mag, 3.0, 7.3)
else:
doresid = False
if doresid:
mmi -= c["C5"] + c["C6"] * ldd + c["C7"] * lmm
pgm = np.zeros_like(mmi)
dpgm_dmmi = np.zeros_like(mmi)
#
# MMI 1 to 2
#
idx = mmi < 2.0
pgm[idx] = np.power(10, (mmi[idx] - c2["C1"]) / c2["C2"])
dpgm_dmmi[idx] = 1.0 / (c2["C2"] * lfact)
#
# Lower segment of bi-linear relationship
#
idx = (mmi >= 2.0) & (mmi < c["T2"])
pgm[idx] = np.power(10, (mmi[idx] - c["C1"]) / c["C2"])
dpgm_dmmi[idx] = 1.0 / (c["C2"] * lfact)
#
# Upper segment of bi-linear relationship
#
idx = mmi >= c["T2"]
pgm[idx] = np.power(10, (mmi[idx] - c["C3"]) / c["C4"])
dpgm_dmmi[idx] = 1.0 / (c["C4"] * lfact)
if imt != self._pgv:
units = 981.0
else:
units = 1.0
pgm /= units
pgm = np.log(pgm)
pgm[ix_nan] = np.nan
dpgm_dmmi[ix_nan] = np.nan
return pgm, dpgm_dmmi
[docs] def getGM2MIsd(self):
"""
Return a dictionary of standard deviations for the ground-motion
to MMI conversion. The keys are the ground motion types.
Returns:
Dictionary of GM to MI sigmas (in MMI units).
"""
return {
self._pga: self._constants[self._pga]["SMMI"],
self._pgv: self._constants[self._pgv]["SMMI"],
self._sa03: self._constants[self._sa03]["SMMI"],
self._sa10: self._constants[self._sa10]["SMMI"],
self._sa30: self._constants[self._sa30]["SMMI"],
}
[docs] def getMI2GMsd(self):
"""
Return a dictionary of standard deviations for the MMI
to ground-motion conversion. The keys are the ground motion
types.
Returns:
Dictionary of MI to GM sigmas (ln(PGM) units).
"""
#
# Need to convert log10 to ln units
#
lfact = np.log(10.0)
return {
self._pga: lfact * self._constants[self._pga]["SPGM"],
self._pgv: lfact * self._constants[self._pgv]["SPGM"],
self._sa03: lfact * self._constants[self._sa03]["SPGM"],
self._sa10: lfact * self._constants[self._sa10]["SPGM"],
self._sa30: lfact * self._constants[self._sa30]["SPGM"],
}
def _getConsts(self, imt):
"""
Helper function to get the constants.
"""
if (
imt != self._pga
and imt != self._pgv
and imt != self._sa03
and imt != self._sa10
and imt != self._sa30
):
raise ValueError("Invalid IMT " + str(imt))
c = self._constants[imt]
c2 = self._constants2[imt]
return (c, c2)
